Stationary max-stable fields associated to negative definite functions

Kabluchko, Zakhar ; Schlather, Martin ; Haan, Laurens de

Document Type: Article
Year of publication: 2009
The title of a journal, publication series: Annals of Probability
Volume: 37
Issue number: 5
Page range: 2042-2065
Place of publication: New York, NY [u.a.]
Publishing house: Inst. of Mathematical Statistics
ISSN: 0091-1798
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematische Statistik (Schlather 2012-)
Subject: 510 Mathematics
Keywords (English): Stationary max-stable processes, Gaussian processes, Poisson point processes, extremes
Abstract: Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi be a Poisson point process on the real line with intensity e−y dy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.

Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation.

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